May/June 2026 Paper 31

A curve has the equation $y={\mathrm{e}}^{\left(x^2+4x\right)}{\cos }^{2}x$, where $\cos x\ne 0$.

Show that the $x$-coordinates of the stationary points of this curve satisfy the equation $\tan x=x+2$.

By sketching a suitable pair of graphs, show that the equation $\tan x=x+2$ has only one root in the interval $-\pi <x<\pi$.

Show by calculation that this root lies between 1.2 and 1.3.

Use the iterative formula $x_{n+1}={\tan }^{-1}(x_n+2)$ to calculate this root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.